absorbing boundary condition for the elastic wave equation: … · 2013-04-01 · absorbing...

GEOPHYSICS, VOL. 54, NO. 9 (SEPTEMBER 1989); P. I141-1152, 9 FIGS., 2 TABLES.

Absorbing boundary condition for the elastic wave equation: Velocity-stress formulation

C. J. Randall*

ABSTRACT

This paper describes an absorbing boundary condition for finite-difference modeling of elastic wave propagation in two and three dimensions. The boundary condition is particularly effective for obliquely incident waves, typically quite troublesome for absorbing boundaries. Analytical predictions of the boundary reflection coefficients of a few percent or less for angles of incidence up to 89” are verified in example finite-difference applications. The algorithm is appropriate for use in a velocity-stress finite-difference (vs-fd) formulation. It is computationally simpler than a similar absorbing boundary given previ- ously for the standard displacement formulation. A second algorithm is presented which may be advanta- geous when the boundary of interest is exposed to strong evanescent waves. Both algorithms require that the adjacent elastic medium be locally homogeneous.

INTRODUCTION

In finite-difference modeling of elastic wave propagation, absorbing boundary conditions are required to prevent spurious reflections arising at edges of the domain of computation from compromising the accuracy of results in regions of interest. In a previous paper (Randall, 1988), henceforth referred to as Ref 1, a new absorbing boundary was introduced for two-dimensional (2-D) and three-dimensional (3-D) finite-difference modeling of elastic wave propagation. This boundary condition, using the notion of scalar potentials, separates compressional and shear components of the incident vector displacement fields, allowing application of an excellent scalar absorbing boundary condition given by Lindman (1975). In Ref 1, I used the standard displacement finite-difference formulation (Kelly et al., 1976). In the present paper, the absorbing boundary condition given in Ref 1 is generalized to the velocity-stress finite-difference

(vs-fd) formulation (Madariaga, 1976; Virieux, 1986). The vs-fd formulation is very flexible and has important advan- tages for modeling irregular fluid-solid interfaces or free surfaces.

In the vs-fd formulation, the new boundary condition is greatly simplified because significant mode conversions at the boundary are eliminated. When a compressional wave is incident, only a (very low-amplitude) compressional wave is reflected. Similarly, only a reflected shear wave results from an incident shear wave. While the waves reaching the boundary are nearly always propagating waves, in the displacement formulation of Ref 1, small but nonzero evanescent mode conversions occur due to finite-difference truncation error. Their presence made necessary the use of the second, computationally more complex version of the scalar boundary condition Lindman presents (henceforth referred to as L2). With the elimination of mode conversions in the vs-fd formulation, Lindman’s simpler scalar boundary (Ll) may be employed, with concomitant reductions in computational expense and programming effort.

While I recommend that Ll be employed whenever pos- sible in the absorbing boundary condition for elastic waves, there are situations in which the absorbing boundary must handle significant evanescent wave energy. For example, interior inhomogeneities may create structures which guide interface waves near the boundary or localized sources may be near the boundary. In such situations, the use of L2 may be preferred. A secondary purpose of this paper is to demonstrate how the implementation of L2 may be consid- erably simplified from that in Ref 1. In particular, L2 employs an operator which has an exact representation only in the wavenumber domain. Fast Fourier transforms (FFTs) are employed in Ref I to make this transformation, a procedure that entails computational expense, restrictions on boundary lengths, and wraparound problems. Here I show that the operator of interest may be approximated directly in the spatial domain by short, computationally inexpensive finite impulse-response (FIR) filters, making the

Manuscript received by the Editor May 23, 1988; revised manuscript received March 9, 1989. *Schlumberger Well Services, P.O. Box 2175. Houston, TX 77252-2175. 0 1989 Society of Exploration Geophysicists. All rights reserved.

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boundary condition more local in nature and eliminating wraparound.

The remaining restriction for application of the new boundary condition, namely, that the medium be locally homogeneous in the immediate neighborhood of the boundary, remains. The elimination of this restriction is the subject of current research.

ABSORBING BOUNDARY FOR ELASTIC WAVES

Consider an isotropic, elastic medium with compressional and shear velocities c and s and density p. The displacement velocity vector v and stress tensor T satisfy the coupled first-order equations

and

; = h(V - v)! + p(Vv + VV), (2)

where I is the identity matrix, V * T is the divergence of a tensor (Jackson, 1962), VV is the transpose of Vv, and where A and p. are the Lame coefficients, p, = ps2 and A + 2~ = pc2. These equations are conveniently solved on a staggered finite-difference grid as shown in Figure 1. Similar staggered grid schemes have also been applied to the time-domain solution of Maxwell’s equations (Taflove and Umashankar, 1987). Virieux (1986) has given a 2-D formulation for elas- ticity. The generalization to three dimensions is straightfor- ward; the finite-difference equations corresponding to equations (1) and (2) are not reproduced here. However, the locations of the various velocity and stress components upon the grid are of interest. Let j, k, and e be the grid indices in the x, y, and z directions, respectively, so that xJ = jAx, y/, = kAy, and ze = [A z, where 0 5 j % J, 0 5 k 5 K, and 0 % 8 5 L. Then the three velocity components are defined as

and

'-'&j~ Yk + 1123 ze + l/2) = vxj,k + l/2,2 + 1127

vybj+ 112, Yk, -3 + l/d = uyj+ lLk,&+ 112,

v&j+ 1123 Yk + 1129 zd = %j + 112,k + 1/2,Y.

The diagonal components of stress are

Tcrct(xj+ l/27 Yk + l/2* Q + 112) = Taaj + 112,k + l/2,( + 112,

where cx stands alternately for x, y, and z, while the nondiagonal components are

%y(xj, Ykt ze + 112) = ‘?xyj,k,e + IL? = 'Ty.x(xj, Yk? Zt + l/2)>

Tyz(xj + l/2$ Yk, ze) = 7yz.j + lL2,k.t = Tzy(Xj + l/29 Yk> ZC)>

and

T.&j, Yk + l/29 Q) = T_rzj,k + l/2,( = Tz.x(xj, Yk + l/i!> Zt).

At the boundary x = x,, equations (1) and (2) fail to produce finite-difference equations for ZI~!,/,~+ ,,2,c+ ,,2, T,,,J,k,e_+ ,,2, or T~~~,~+,,*,~ due to the truncation of the finite-difference grid.

The function of the absorbing boundary is to provide these three quantities or equivalents such that spurious reflections are minimized.

If the elastic medium is locally homogeneous near the boundary of interest and the stress 7 eliminated, equations (I) and (2) combine to produce the vector wave equation

$V - c%(V * v) - s2v x v x v.

2- (3)

The velocity may be written as a sum of curl-free and divergence-free parts

v=V$+VkA, (4)

:- j=O

FIG. 1. Finite-difference grid for 3-D vs-fd calculations. The 3-D staggered grid is depicted as an alternating stack of 2-D y - z planes. The absorbing boundary condition is applied at x,. The velocity vX is known on the integral j planes shown in bold outline, and the transverse velocities v and v, .are known on the half-integral j planes shown in Tight outhne. The indices k,t mentioned in the text run along the y and z directions, respectively, with integral k,t lines shown in bold on the integral j planes, and shown in light lines on the half-integral j planes. The labeled symbols define the locations of the various field quantities within the grid.

Absorbing Boundary Condition 1143

where + is the scalar (compressional) potential and A is the vector (shear) potential. The specification of the vector potential may be made unique through the auxiliary condition V 6 A = 0. The potentials satisfy the wave equations

a24 2 = c2v24 (5)

and

a2A ~=-s~V.VXA=~~V~A. (6)

In view of equation (4), the right members of these equations may be written in terms of the velocity as

a24 -c2v*v z- and

a2A -$ = -s2v x v.

The compressional potential + and the three Cartesian components of the shear potential A satisfy uncoupled scalar wave equations and, therefore, may be individually treated by an absorbing boundary condition for scalar waves. This suggests the following three-step absorbing boundary for elastic waves: First, equations (7) and (8) are used to express the vector velocity field in terms of the four intermediary uncoupled scalar potentials + and the three Cartesian components of A. Next, a scalar absorbing boundary, either Lindman’s Ll or L2, is applied to these potentials; and finally, equation (4) is employed to recover the vector velocity field from the potentials. Note that this scheme does not provide new shear stresses rXy or rXZ on the boundary x,, but, equivalently, does provide the transverse velocities 7~~~

and v, just interior to the boundary x,_ ,,2. In Lindman’s paper, both boundary conditions Ll and L2

are given in terms of difference equations in two dimensions. In Ref 1, corresponding differential equations are given, allowing easy generalization to three dimensions and application to alternate differencing schemes. For convenience these equations are given here. Boundary condition Ll may be written

;++

m=M

1 h,, WI=1

where

a'h, ,,z - Pmc’V; h, = a m ~‘0: (10)

and where

a2 a’ VZ=g+g (I 1)

is the transverse Laplacian operator. Ll is effective for propagating scalar waves and is slightly more complex than paraxial boundaries (Clayton and Engquist, 1977; Engquist and Majda, 1979). The coefficients (Y, and B, are obtained

by minimizing a weighted average of the reflection coefficients. With just three terms (M = 3) in the series, Lindman obtained reflection coefficients less than 0.01 for propagating waves incident at any angle up to 89”. The coefficients for M = 3 are (Y, = 0.3264, 0.1272, 0.0309 and B,,, = 0.7375, 0.98384, 0.9996472.

The second boundary condition L2 accurately treats evanescent as well as propagating waves but is computationally more complex. It may be written

a4 X+C

where

a2h, afbrl ,,z + qmK t - c2V$,

z ~‘0: + (1 - y&K ; (13)

The coefficients a, and Y,,, are also obtained by minimizing a weighted average of the reflection coefficients, although the average now includes evanescent waves. Excellent ab- sorption is obtained with M = 6. The coefficients are CI,,, = 0.5155, 0.2723, 0.1232, 0.05541, 0.02333, 0.01030 and yrn = 1.6543, 0.4922, 0.09891, 0.01637, 0.002062, 0.0001395. The reflection coefficient is less than 0.01 for propagating waves incident at any angle, except in the immediate neighborhood of 90” and, in addition, is correspondingly small for evanescent waves. The operator K( . ) denotes transformation into the transverse wavenumber domain, multiplication by the magnitude of the transverse wavenumber Ik,l = (k,z + kz)“‘, and inverse transformation back to the space domain. Formally, K may be written as the convolution

dy’ dz' fly', z’)

&, &_ ,k I , &(J’ - Y’) + & - ?‘)I. L (14)

Both Ll and L2 are based upon best-fit rational series expansions of the one-way wave equation (projection operator) in terms of the transverse wavenumber, whereas paraxial boundaries employ formal power or Pade series. The rational expansion is better able to approximate the singular behavior of the projection operator throughout the entire range of incidence angles. Stable, second-order accurate finite-difference implementations of the new absorbing boundary condition using both Ll and L2 have been obtained and are described in Appendix A. In the next section, numerical results are given which demonstrate with both numerical evaluation of the reflectivity matrix and example finite-difference calculations the effectiveness of the new absorbing boundary condition.

NUMERICAL RESULTS

The absorbing boundary condition is formulated in the space-time domain; reflectivities are properly considered in

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the wavenumber-frequency domain. The reflectivity matrix R relates incident and reflected waves as a function of frequency, polarization, and transverse wavenumber. Inci- dence angle is related to wavenumber through 0 = sin-’ (ck,/w), where c is the wave propagation speed and o is angular frequency. For evanescent waves, where ck, > w, 0 is complex. The determination of R for the present boundary conditions closely parallels thatin Ref 1. Details of the calculation are given in Appendix B.

In Figure 2, the diagonal elements of the reflectivity matrix

8 a

3

0.

-1.

-2.

-3.

%

(a)

0. 16. 30. 46. 60. 76. 90

0s

@I

FIG. 2. Diagonal elements of absorbing boundary reflectivity matrix in two dimensions including finite-difference error. Parameters are c = l., s = 0.5, Ax = Ay = 0.15, and At = 0.075. Curves 1 and 2 are for w = 1 and w = 2. (a) Common logarithm of R,, versus compressional incidence angle. (b) Common logarithm of R,, versus shear incidence angle.

are displayed for the elastic absorbing boundary condition employing Ll. The nondiagonal elements, which represent mode conversions at the boundary, are three to four orders of magnitude smaller, and may be the result of finite numerical precision. The compressional and shear speeds are c = 1 and s = 0.5 and the finite-difference grid parameters are AX = Ay = 0.15 and At = 0.075. For simplicity, the propagation is 2-D: k; = 0. Results are given for two frequencies, w = I and o = 2. For w = 1 (about 40 and 20 grid points per compressional and shear wavelength, respectively), finite-difference error is moderate, while for o = 2 (about 20 and 10 grid points per wavelength), finite-difference error is substantial. The common logarithm of lR,.,,i, the reflectivity for an incident compressional wave reflected as a compressional wave, is displayed in Figure 2a versus compressional wave incidence angle 0, = sin’ (ck,,h),

measured in degrees. For both frequencies, the reflectivity remains below 0.01 for angles of incidence up to 89”. At exactly grazing incidence, 8, = 90”, the reflectivity is unity. The two shear modes, I and 2 (see Appendix B), have identical reflection coefficients R,, = Rz2 = R,,,y. Since shear waves are relatively less well represented on the finite- difference grid than are compressional waves at a given frequency, the shear-shear reflectivity lR,_ shown in Figure 2b is more sensitive to finite-difference error. The reflectivity at the higher frequency is a few percent at normal incidence, but like IRJ, does not become significant until grazing incidence, tl, 2 89”. The reflectivities displayed in Figure 2 show more than an order of magnitude improvement for oblique incidence compared to those from paraxial boundaries in common use (Ref 1).

In Figure 3, the three diagonal reflectivities are displayed for full 3-D propagation, again with the Ll scalar boundary. All parameters are the same as in the previous example

0.0 0.6 1.0 1.6

FIG. 3. Diagonal elements of absorbing boundary reflectivity matrix in three dimensions including finite-difference error. Parameters are c = l., s = 0.577, AX = Ay = 0.15, and At = 0.075, w = 1. Curve labeled CC is the common logarithm of IR,., I. Curve labeled 1 I,22 is the common logarithm of IR, ,I = lR,21.


except that s = 0.577 and only results for the lower frequency, w = 1, are shown. In this figure, the common logarithms of the reflectivities are plotted against the transverse wavenumber k,. A constant ratio of GIG = 1.5 is maintained as k, is varied. The label CC in the figure stands for the compressional normal mode; and the labels 11 and 22, for the two polarizations of shear modes. The reflection coefficients for this 3-D case are very similar to the 2-D results of Figure 2. The transformation from 8 to k, has compressed the high 8 regions on the abscissa. The scalar boundary condition Ll does not account for evanescent waves, and the reflection coefficients are large in the evanescent regions k, 2 o/c = 1 for the compressional mode and k, 2 o/s = 1.73 for the shear modes. In the absence of finite-difference error, the Ll reflection coefficients are exactly unity in the evanescent region. In practice they may become slightly larger, reaching about 1.2 in this case for the largest k,. However, this slight amplitication is insignificant compared to the very strong decay both toward and away from the boundary for the evanescent waves. The nondiagonal reflectivities for the 3-D case are also insignificant.

In Figure 4, contour plot snapshots from 2-D vs-fd calculations are displayed. The region of interest comprises two elastic half-spaces in welded contact. In the lower half- space, c= 1, s = 0.577, and p = 1, while in the upper half-space, c = 0.75, s = 0.4, and p = 0.6. A point force in the x direction is applied at the origin with a time dependence in the form of a wide-band, zero-mean pulse (first derivative of a Blackman window.) The center frequency of the timepulse is 0, = 0.75. The finite-difference parameters are Ax = Ay = 0.2 and At = 0.1. In Figure 4a, the new absorbing boundary condition using Ll is applied at the upper and lower horizontal boundaries y = 220. Antisymmetry is invoked at the left boundary, while a second-order paraxial scheme is applied at the right. The time of this plot is t = 74. Many clusters of contour lines are evident, each representing a propagating wavelet. The green lines are contours of IV - VI (compressional wave fields) while the red lines are contours of IV x VI (shear wave fields). In the lower half-space, the compressional body wave is found with its leading edge at x = 74, while in the upper, slower half-space, the body wave is found at x = 55. A head wave, itself a compressional disturbance, arises at the interface in the upper half-space and connects the two compressional body waves. Two additional head waves, both shear disturbances, connect the lower compressional body wave at the interface with the shear body waves found at x = 43 and x = 30 in the lower and upper half-spaces, respectively. Three more head waves complete the picture. Two of low amplitude, one in each half-space, are generated by the upper compressional body wave at the interface; and one in the upper half-space is generated by the lower shear body wave. Evanescent

X (a)

X W

FIG. 4. Contours of IV - VI (compressional wave field in green) and IV x VI (shear wave field in red) for 2-D finite-difference calculations of two elastic half-spaces in welded contact. A horizontal force point source is located at the origin. Lowest contour level is 1 percent of the maximum amplitude. (a) The new absorbing boundary condition is applied along upper and lower boundaries. (b) A second- order paraxial absorbing boundary condition is applied.

1146 Randall

compressional wavelets on the interface are visible at the lower shear body wave, but are too small to be visible near either of the upper body waves. The lowest level contour lines are at 1 percent of the maximum of the plotted function, with amplitudes normalized separately in the upper and lower half-spaces.

The effective angles of incidence of the two compressional body wavefronts with respect to the absorbing boundaries in Figure 4a are about 70” and 75”, yet, consistent with the reflection coefficient calculations, reflections and mode conversions at the boundaries are negligible. The two shear body waves are incident upon the boundaries at about 56 and 65”. No mode conversions to compressional are evident. Reflected shear wavelets traveling toward y = 0 are visible. Consistent with Figure 2b, these wavelets are dominated by high-frequency components: they are maximum for normal incidence; but most importantly, their amplitude is on the order of 1 percent. These small reflections are due to truncation error (grid dispersion) in the boundary finite- difference formulation. The high-frequency ripple with an amplitude about 1-2 percent trailing the shear body wave is also a manifestation of grid dispersion but on the interior of the grid. Both effects could be reduced by refining the grid. With the effective angle of incidence of the body waves so close to grazing, this calculation represents a severe test of an absorbing boundary condition. In Figure 4b, a second- order paraxial boundary is applied at the upper and lower horizontal boundaries. The compressional body waves in both half-spaces are distorted by reflections from the boundaries, and shear pseudo-head waves have been generated. Similarly, reflections of the shear body waves at the boundaries and mode conversions into evanescent compressional waves are evident.

In another vs-fd calculation, the upper half-space is made identical to the lower one so that only two body waves are excited: a compressional wave which is a maximum on the plane y = 0 and a shear wave which is maximum on the plane x = 0. In Figure 5, the total kinetic energy, labeled T, is plotted versus time T is the integral over the region 0 I x 5 75, lyl 5 20 of the quantity Iv,I’, where v, is the velocity obtained for a very large (effectively infinite) grid. After an initial rise, T is constant until f = 20, when compressional energy reaches and begins to pass through the upper and lower integration boundaries. At t = 35, T falls more steeply as shear energy reaches these boundaries; and at t = 75 and t = 130, Tfalls further as the compressional and shear waves reach the right integration boundary. The three curves labeled 1, 2, and L are residual energies. Each is a spatial integral of 16v12, where 6v = v - v, and where v is the velocity obtained on a smaller grid with absorbing boundaries at y = +20 and x = 75. These curves measure the energy reflected by the (imperfectly) absorbing boundaries. Curves 1 and 2 result when first- and second-order paraxial boundaries are applied. The second-order boundary is about a factor of three better than the first-order boundary in this example, but the new absorbing boundary, curve L, yields an additional order of magnitude improvement. (The adjacent medium is now homogeneous, so that the new boundary condition can be applied on the right.) Since the waves pass completely out of the finite-difference grid, this calculation also exercises the corner point conditions (see Appendix A).

Numerical results up to this point have been for the new absorbing boundary condition using Ll. In Figure 6a, the diagonal components of the reflection matrix when L2 is used (solid curves) are compared with those obtained when Ll is used (dashed curves). The propagation is 2-D, w = 1, and the parameters are those of Figure 2, except that the abscissa is transverse wavenumber instead of incidence angle. The two curves near the label CC are the compressional reflection coefficients. The Ll curve (dashed) remains at lR,.,.l = 1 in the evanescent region k, 2 1, while the L2 curve (solid) has a narrow spike at exactly grazing incidence and then falls back below 0.01 again in the evanescent region. Similar results are obtained for the shear reflections (labeled 11,22). The Ll curve (dashed) remains at approxi- mately 1 throughout the evanescent region for shear, k, 2 2, while the L2 curve immediately drops back down below 0.01 as k, enters the evanescent region. The price paid for the more general applicability is significant computational com- plexity. The finite-difference implementation of the boundary condition L2 involves an FIR filter (see Appendix C). The L2 results in Figure 6a are for an infinite filter length, essentially equivalent to the FFT technique of Ref 1.

The filter length may be reduced to decrease computation time and make the boundary more local in nature. In Figure 6b, the L2 reflection coefficients are displayed for several length 21 filters [KL = 10 in equation (C-2)]. The solid lines, curves (a) and (c), are I&. and lR,,I, respectively, when the filter proposed by Lindman, a Shepp-Logan filter (Shepp and Logan, 1974), is used with KL = 10. The dashed curves, (b) and (d), are &.,.I and IR,,l, respectively, when a modified set

FIG. 5. Kinetic energies within the region 0 5 x 5 75, lyl 5 20 versus time for finite-difference calculations similar to those of Figure 4 except that the upper and lower half-spaces are identical. Curve T is total kinetic energy when boundaries are effectively infinitely far away. Curves 1 and 2 are residual kinetic energies (energy of reflections) when first- and second-order paraxial boundaries are employed at upper, lower, and right boundaries. Curve L is residual kinetic energy when new absorbing boundary (Ll) is employed at upper, lower, and right boundaries.


___-_--

kl (a)

0.0 0.6 1.0 1.6 2.0 2.6

kl (W

FIG. 6. Diagonal elements of absorbing boundary reflectivity matrix in two dimensions including finite-difference error. Parameters are c = 1, s = 0.5, Ax = Ay = 0.15, and At = 0.075, w = 1. (a) Comparison of L2 boundary for filter half-lengths KL + m (solid curves) with Ll boundary (dashed curves). Curves labeled CC are the common logarithm of I&.. Curves labeled 11,22 are the common logarithm of IR,,I = IR,,I. (b) Comparison of L2 boundaries for filter half-length KL = 10. Solid curves are with Shepp-Logan FIR filter; dashed curves are with modified filter. Curves a and b are IR,.,I and curves c and d are IR, ,I = IR,,I.

0.6

5=

52 II 0.0 25 >”

-0.6

FIG. 7. Horizontal velocity versus time at the position x = 50 for a finite-difference calculation similar to that of Figure 4 except that p = 0 in the upper half-space, so y = 0 is a traction-free surface. The two solid curves result when L2 is applied at y = - 10 and when the boundary is far removed, so that boundary reflections are absent. The dashed curve results when Ll is applied at y = -10.

of filter coefficients is used. The modified filter has been designed to reduce the reflections near normal incidence using a linear programming algorithm (Steglitz and Parks, 1986). The filter coefficients are given in Table C-l. Either of the short filters gives acceptable results, although the modified one yields a broader range of k, over which IR,.,.I and IR,,yl are less than 0.01.

An example in which the more general boundary using L2 yields improvement is shown in Figure 7. Displayed are horizontal velocity waveforms at (x, y) = (50, 0) for a calculation similar to that of Figure 4a, except that the upper medium has p = 0, so that the plane y = 0 is traction-free and a strong Rayleigh wave is present. The lower boundary is moved in to y = - 10. The solid curves in Figure 7 are the waveform obtained using L2 (modified FIR filter, K, = 10) and the waveform with the lower boundary far removed so that boundary reflections are absent. Good agreement is obtained. The dotted curve is obtained when the Ll boundary is used. The passage of the Rayleigh wave, in particular, exposes the boundary to strong evanescent compressional and shear components. Since the Ll boundary does not handle these components properly, departures from the correct result occur. However, if the boundary is moved back to y = -20 (several Rayleigh wavelengths), the magnitude of the evanescent components at the boundary is substantially reduced; and the Ll and L2 results agree.

CONCLUSION

A new absorbing boundary condition has been given for the velocity-stress formulation of the finite-difference elastic wave equation. The new algorithm uses the notion of scalar potentials to separate compressional and shear components

1148 Randall

of the incident vector field and then applies one of the elegant schemes (called Ll and L2 here) given by Lindman to these scalar components. The version of the new boundary which employs Lindman’s Ll scheme is ebier to imple- ment and is widely applicable. The boundary employing L2 is formulated in terms of an FIR filter and offers the occasionally important advantage of correctly handling evanescent waves. Both versions of the new boundary are much more absorptive than paraxial absorbing boundaries in common use. For propagating waves, boundary reflectivities are less than a few percent for oblique incidence up to 89”.

Synthetic seismograms: a finite-difference approach: Geophysics, Al 7-37 . _, - _

Liao, Z. P., Wong, H. L., Yang, B. P., and Yuan, Y. F., 1984,. A transmitting boundary for transient wave analysis: Scientia Simca A. 27. 1063-1976.

Lindman, E. L., 1975, Free space boundaries for the scalar wave equation: J. Comp. Phys., 18, 66-78.

Madariaga, R., 1976, Dynamics of an expanding circular fault: Bull. Seis. Sot. Am., 66, 163-182.

McClellan, .I. H., 1973, The design of two-dimensional digital filters by transformations: Proc. 7th Ann. Princeton Conf. Information Sciences and Systems, 247-251.

McClellan, J. H., and Chan, D. S. K., 1977, A 2-D FIR filter structure derived from the Chebyshev recursion: IEEE Trans. Circuits and Systems, 24, 372-378.

Randall, C. J., 1988, Absorbing boundary condition for the elastic wave equation: Geophysics, 53, 61 l-624.

Shepp, L. A., and Logan, B. F., 1974, The Fourier reconstruction of a head section: IEEE Trans. Nucl. Sci.. NS-21, 21-43.

Steglitz, K., and Parks, T. W., 1986, “What is the filter design problem”: Proc. 1986 Conf. on Information Science and Systems, 604-609.

REFERENCES

Clayton, R., and Engquist, B., 1977, Absorbing boundary conditions for acoustic and elastic wave equations: Bull. Seis. Sot. Am., 67, 152%1540.

Engquist, B., and Majda! A., 1979, Radiation boundary conditions for acoustic and elastic wave calculations: Comm. Pure Appl. Math., 32, 314-357.

Jackson, J. D., 1962, Classical electrodynamics: Wiley. Kelly, K. R., Ward, R. W., Treitel, S., and Alford, R. M., 1976,

Taflove, A., and Umashankar, K. R., 1987, The finite-difference time-domain (FD-TD) method in electromagnetic scattering and interaction problems: J. Electromag. Waves and Appl., 1, 243- 267.

Virieux, J., 1986, P-SV wave propagation in heterogeneous media: Velocity-stress finite-difference method: Geophysics, 51, 889- 1001.

APPENDIX A

FINITE-DIFFERENCE IMPLEMENTATION

The scalar potential + is defined centered one-half cell interior and one-half cell exterior to the boundary plane x = x,. A, is defined centered one-half cell interior to the boundary plane, while the potentials A, and A, are defined centered one full cell interior to and on the boundary plane. For each potential, the interior location is advanced in timethrough finite-difference approximations to equations (7) and

W$ - 1,k.e + 112

At2

2 Q&

=-SC

3l2,k.e + l/2 $?:J - I k - 112 e + , . l/2 _

Ax 1 AY ’ (A-4)

where 6, denotes the forward difference operator in the x(j) direction, S,f, = J;+, - jj, and 6,gj-,,2 = gj+,,2 - gj-,,2. Difference operators in the y and z directions are similarly defined. In equations (A-I)-(A-4), the time index II has been introduced, with t” = nAt, and S = f(P). The forward difference operator in the time direction is S,f” = S” - S, while the second central difference operator is SJ“ = S” - 2f” + f”-‘. At the beginning of each new time step, t” -+ tn*‘, these equations are solved for the appropriate time- advanced interior scalar potential ~$$Jnf:,~,~+,,~,~+ ,,* in, for example, equation (A-l).

In order to reconstruct from equation (4) time-advanced velocities on the boundary, x derivatives of the potentials + “+I, AJ!+‘, and A:+’ are required. These derivatives are approximated by differences of exterior and interior values across the boundary. The interior values have been obtained in equations (A-I)-(A-4). The exterior values are obtained through application of the scalar absorbing boundary

M+; + l/2 + +I; - 112) u+lf : ;,2 + 44 - 112)

2At + cq

2Ax m=M

=- n + l/2. c %fz T (A-5)

(8):

h4: - 1/2,k + l/2,( + 112

At2

I,k + l12,e + 112 6 un + Y yJ - 1/2,k,t + Ii2

AY

+ 6zv.iJ - l/2,k + 112,t

AZ (A-l 1

= _S2 6,u~~l12,k-1/2,~ 6zv;J-l/2kl-1/2

Ay - ” AZ ; 64-2)

%,A;J - l,k + 1/2,e

At2

= -s2~szv~J -

1,k + 112.t - l/2 sx”; - 3/2,k + Il2,e _

AZ Ax 1:

(A-3)

m=l

W;J + A_:_, - 1) MA;;+_‘, + A;J - I)

2At + sq 2Ax

m=M =- c h;; li2;

m=l

(‘4-6) and


and

%(A:, + A> - ,) S&4$+_‘, + AyJ _ ,)

2At + sq

2k $$J:,; + 1/2 ?&A n+l

+ yJ.h + 1:2x.

Ay - AZ ’ (A-l 1)

(A-7) n+l WJ” T II/?& - 112.1 + l/2

7’yJ I:?,k.Y + Ii2 = AY

where q = 1 for the Ll scalar boundary and

q= cc+* = 1 +

f&C/+-‘I/W S&r+-‘1,k.t + 112.

AZ - AX , (A-12)

for L2. The k and e indices have been suppressed for clarity. Equations (A-5)-(A-7) are solved for the time-advanced exterior potentials 4.7: :,2, A$ ’ , and A:Jt ’ , respectively. The time-advanced correction functions h ;zt”* in equation (A-5) are obtained from the finite-difference equation

and

II + I

n + I s4.J li2.k + I/?,[ - I!?

7’3 - l/2,/, t li2.Y = AZ

S,, h&j “2 srw - I /2 + 4); :;,:I At2 - Pmc2V;dh;,; “’ = CX,,,&‘:~

24x

(A-8)

for boundary condition Ll , or from

S,,h;, 1’2 Ar’

= (Y c3v2 fd+; - l/2 + 44 1 I/?)

m Id 2Ax

+ cdl - yrn)c2K~

fs,+l; I 112

AtAx 64-9)

for L2. Here K, is a discrete representation of the operator K( - ) discussed in Appendix C, and V:,, is the discrete representation of the transverse Laplacian

(A-IO)

Analogous equations apply for h$‘12 and hg,:“’ with s replacing c and A, or A, replacing $.

Finally, time-advanced velocities v”+’ on the boundary are obtained through a finite-difference approximation of equation (4) applied to the time-advanced potentials:

Equations (A- I)-(A- 13) comprise the finite-difference em- bodiment of the new absorbing boundary conditions for elastic waves in the velocity-stress formulation. All of the equations are explicit with the exception of equation (A-9) for the L2 correction functions, in which K, operates on h $Ll”‘, coupling adjacent boundary points. Instead of invert- ing the resulting matrix, an iterative approach can be adopted. Iteration converges quite rapidly, since the matrix is very much diagonally dominant. For the LI boundary, the equation for the correction functions, equation (A-8), is explicit.

At the edges and vertices of a 3-D grid, and at the corners of a 2-D grid, some of the potentials required in equations (A-l I)-(A- 13) are not available. However, satisfactory results are obtained when approximate extrapolated values are employed for these corner point quantities. The extrapolation uses values at neighboring grid points on the boundary. Any one of many schemes may be used. For the 2-D finite-difference results presented here, a third-order “multitransmitting” extrapolation scheme (Liao et al., 1984) is applied to generate corner values for both the potentials and the correction functions. For the Ll boundary condition, a single extrapolated value is required at each corner for each correction function, while for the L2 boundary, many values are required, depending on the length of K,. These values may be generated by successive application of the extrapolation scheme.

APPENDIX B

REFLECTIVITY MATRIX R

In this appendix the reflectivity matrix of the new absorbing boundary condition is calculated. Transforming to the wavenumber-frequency domain and defining the interior and exterior potential vectors 6inl = ($,_,,,, A,,_ r12, A,,_, , A,_,) and &‘,,t = (&J+ ,,2, A,_,,,, &, A,) and the interior and exterior velocity vectors Gint = ($_,, 5’yJ_-3,2, tizJ_3,z) and B,,, = (a,, tiY,_1,2, IL’+ ,,2), equations (A-I)-(A-4), (A-&o-(-7), and (A-ll)-(A-13) may be written in compact form

and

(B-1)

(B-2)

(B-3)

1150 Randall

where I is the identity matrix and the tilde above a vector denotes a transformed quantity. A and B are the 4 x 3 matrices

42 S:

[

-pf 0 0

0 - ipzSz

00 1 0 - P:

(B-4)

i&S, PZ 0

and

[

p: iPfsy 0 iP~Si

0 0

0 -PZ

@fS,

- iP3,

PS ’ 0 1 (B-5)

where for simplicity Ax = Ay = AZ = A, while Bc = cAtlA, B, = sAtlA, S,,, = 2 sin (kY,ZA/2), and S, = 2 sin (wAti2). p and E are the 3 X 4 matrices

D -[ = $

-10 0 0 ‘S>> 0 0 1 1 (B-6) ISi 0 - 1 0

and

F=$ j;; -F 411. (B-7)

c is a 4 x 4 diagonal matrix with C, , = C,., C,, = 0, and C,, = C, = C,, where

(B-g)

and where for LI

(B-9)

while for L2

i T + Zi&SS,[l - 2~,~ sin’ (wAt/4)].

m = , S; + 2iy,&BS, cos (oAt/2) - p,,S:

(B-IO)

In these equations, IT stands for either c or s, and ST = S: + Sf. The spectral response B of the operator K, is defined in Appendix C.

Combining equations (B-lj(B-3) yields

where

+ext = (! + I;)%* (B-l 1)

q = [r - (p + E(1 + C))B] - ‘[(Ij + E(1 + C))(+ + B) - I]. _ _ _ _ (B-12)

The velocity field + is decomposed into incident (+) and reflected (-) parts:

i=*+ +q-. (B-13)

Using equations (B-l I) and (B-13), a reflectivity matrix may be formulated which relates incident and reflected Cartesian velocity components. However, the reflection coefficient is more readily interpretable when formulated in terms of the normal modes of the finite-difference equations. The transformation between Cartesian components and normal modes is derived next.

The vs-fd equations corresponding to equation (3) reduce to the homogeneous linear system

WO=O. (B-14)

A typical diagonal component of W is given by w, = pf$ + p(s; ; s;, - s:, (B-15)

while a typical nondiagonal component is given by

w,, = w,, = (P: - P;)S.YS,* (B- 16)

where S, = 2 sin (k,A/2). The dispersion relation is

det (W) = (Sf + S: - s:/pf)(s; + s: - Sf&)’ = 0, (B-17)

yielding two pairs of solutions for S,. One pair, S,, = *(s;/pf - Sz_)“2, corresponds to right-going and left-going (+) and (-) components of a compressional normal mode cc. The other pair, S,,, = t(Sf/Bz -S:)“‘, corresponds to right-going and left-going components of either of two shear normal modes C, and V2. The compressional and shear wavenumbers are kl( = 21A sin-’ (S,,.) and k,, = 2/A sin’ (S,,). The velocity vector of the compressional normal mode is nearly parallel to the propagation vector k = (km ky, kJ:

?,’ = (+S,,% + $3 + S&A-‘. (B-18)

The two orthogonal polarizations of shear modes have velocity vectors perpendicular to k:

and

-+ _ ( St& &‘G ~1

v2- 8,sI-p,s.” A ) . (B-20)

The Cartesian components of the normal modes are deter- mined by equation (B-14) to within an arbitrary normaliza- tion constant. The magnitude of the compressional mode has been normalized to k,. and that of the shear modes, to k,.

The incident and reflected velocity fields may be expressed in terms of the normal modes. Let the transformations be denoted by

+’ = M++, (B-21)

where 9’ are the incident and reflected velocities, using as basis vectors the finite-difference normal modes, and the M, are transformation matrices. The transformation matrices l’$, are formed from the column vectors representing the components of the three normal modes

PI/I, = {?f, SF, St}-‘. (B-22)

The normal modes propagate in the x direction as


(B-23) (B-25)

where K is a diagonal matrix whose nonzero elements K, are the roots of the finite-difference dispersion relation, equation (R-17), Kjj = (k,,, k,,, k_J.

Using equations (B-21) and (B-23), the incident and re- fleeted velocities at the interior and exterior positions are

Substitution of equations (B-13) and (B-24)-(B-25) into equation (B-l 1) yields

?,, = Rq,tt,

where the reflectivity matrix R is given by

(B-26)

expressed in terms of the normal modes: -i vext = M; ‘9’ - ext

and

(B-24) Ip = [(! + M GM I ‘)ciKa - I] -- ‘[hj _ bj + ‘1 _ __

APPENDIX C

FIR FILTER REPRESENTATION OF K( . )

““J pm J -cc

Lindman approximates the integrals as discrete Fourier transforms and replaces Ik,l by lk,i sine (k,Ay/2). The resulting discrete expression is ’

C-2)

(C-3)

where

q, 2 l , 7~ 1 - 4k’-

x [I - (I + i’yl+ GM; ‘)e jKa] _ _ - _

It is instructive to first consider a 2-D case, a/& = 0, for which the operator K(e) reduces to

Kf(y) = & c co dk, lkYl I- = dy’ &‘py’)f(y’). (C-l)

2.6

2.0

1.6

b

1.0

0.6

0.0

FIG. C-l. FIR filter spectral response. The solid curve is the Shepp-Logan filter with KL -+ =; the dashed curve (almost coincident with the solid curve) is the Shepp-Logan filter with KL = 10; and the dotted curve is the modified filter with K, = 10.

(B-27)

in the limit of a long DFT. This same FIR (finite impulse response) filter appears frequently in the tomography litera- ture, where it is called the Shepp-Logan filter (Shepp and Logan, 1974). The length of the filter is L, = 2K, + 1.

0.0 , ( , , , , , , I , I, I I, I I I I

-0.60 -0.26 0.00 0.26 0

P

FIG. C-2. FIR filter spectral response. Same as Figure C-l except on expanded scale.

Table C-l. Coefficients of the modified one-dimensional FIR filter for K,> = 10.

k

0 1 2

:

:, 7 8 9

10

1.20160 -0.36869 -0.11304 -0.02473 -0.01453 -0.03151

0.03041 -0.07303

0.04572 0.00788

-0.04044

1152 Randall

Table C-2. Coefficients of a two-dimensional FIR filter generated from the one-dimensional modified filter of Table C-l using the McClellan transformation. The coefficient matrix is symmetric; only half of the coefficients are shown.

k’ - T/c,/,

0 1.68399 -0.17344 -0.01986 -0.00579 -0.00260 -0.00121 -0.00088 -0.00040 -0.00050 -0.00014 -0.00056 ; -0.09881 -0.01553 -0.00823 -0.00525 -0.00361 -0.00186 -0.00225 -0.00111 -0.00129 -0.00071 -0.00059 -0.00051 -0.00058 -0.00038 -0.00022 -0.00023 -0.00040 -0.00038 -0.00051

: -0.00223 -0.00143 -0.00084 -0.00076 -0.00073 -0.00068 -0.00051 -0.00038 -0.00024 -0.00026 -0.00042 -0.00048 -0.00040 -0.00023 -0.00012 :, -0.00056 -0.00027 -0.00035 -0.00036 -0.00046 -0.00046 -0.00032 -0.00024 -0.00011 -0.00001 -0.00005

7 -0.00034 -0.00015 -0.00003 -0.00000

; -0.00004 -0.00001 -0.00000 -0.00000 -0.00000 10 -0.00000

The spectral response of the filter is obtained by substi- tuting for f, the exponential function jjj = exp (ikp), where p = k,Ay, and k, is the wavenumber

$(p) = AJ, y zz

KI.

To + 2 2 w cos VP). (C-4) X’ = ,

In Figures C-l and C-2, this spectral response is displayed for the Shepp-Logan filter for K,_ = 10 (dashed curve) and for KL + 30 (solid curve). The dotted curve is a modified filter discussed below. In Figure C-l the entire range of --n s p 5 T is shown. On this scale, the effect of finite K, is not visible. In a finite-difference calculation, however, only the range -0.5 5 p 5 0.5 is represented accurately on the grid. In Figure C-2 the spectral responses are plotted on this expanded scale. The finite length filter response oscil- lates about the desired infinite filter response and does not reproduce the sharp cusp near p = 0. When implemented in a boundary condition, this error introduces an additional reflectivity near normal incidence, which actually increases as the finite-difference grid is refined if K, is fixed. This error

may be exchanged for increased error for large lpl by modifying the filter coefficients r),. The coefficients in Table C- 1 were obtained through a linear programming approach to filter design (Steglitz and Parks, 1986). The spectral response of this filter is shown in Figures C-l and C-2 as the dotted curves. The exaggerated errors for I pl 2 1 in Figure C-l are of no practical consequence, since such waves are not accurately represented on the grid in any case. As long as 3(p) 5 2, numerical stability is not affected. In Figure C-2 the improved response near I pl = 0 is evident. For 3-D calculations, a 2-D FIR filter is required:

&(.fd = 2 (C-5) h’ = pK[, Y’ = -KI

The McClellan transformation (McClellan, 1973; McClellan and Chan, 1977) may be applied to generate these filter coefficients from any satisfactory one-dimensional set. The coefficients displayed in Table C-2 are the result of this procedure applied to the modified 1-D filter of Table C-l.

absorbing boundary condition for the elastic wave equation: … · 2013-04-01 · absorbing...

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